#239033

order	512
nilpotency class	3
isoclinism	isoclinic class of #239025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7)(3,11)(4,5)(8,15)(9,10)(12,16)(17,18)(19,27)(20,21)(22,26)(23,25)(24,29)(28,31)(30,32);(1,2,6,10)(3,12,11,15)(4,9,5,7)(8,13,16,14)(17,19,23,31)(18,24,25,32)(20,27,26,28)(21,30,22,29);(1,3,4,13,6,11,5,14)(2,8,7,15,10,16,9,12)(17,20,21,18,23,26,22,25)(19,28,29,24,31,27,30,32);(1,4,6,5)(2,9,10,7)(3,13,11,14)(8,12,16,15)(18,25)(20,26)(24,32)(27,28)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146229);